Question

Let \(\overrightarrow{a}=3\^i+\^j −\^k\)  and \(\overrightarrow{c}=2\^i+3\^j −3\^k\)  if \(\overrightarrow{b}\) is a vector such that  \(\overrightarrow{a}=\overrightarrow{b}\times\overrightarrow{c}\) and \(|\overrightarrow{b}|=50\) , then \(|72|\overrightarrow{b}+\overrightarrow{c}|^2|\)| |  is equal to _______. 

Answer 1

Correct Answer: 66

Given Vectors:

\(\vec{a} = 3\hat{i} + \hat{j} - \hat{k}, \quad \vec{c} = 2\hat{i} - 3\hat{j} + 3\hat{k}.\)

Step 1: Magnitude of \( \vec{a} \) and \( \vec{c} \)

\(|\vec{a}| = \sqrt{3^2 + 1^2 + (-1)^2} = \sqrt{9 + 1 + 1} = \sqrt{11},\)

\(|\vec{c}| = \sqrt{2^2 + (-3)^2 + 3^2} = \sqrt{4 + 9 + 9} = \sqrt{22}.\)

Step 2: Relationship between \( \vec{a}, \vec{b}, \) and \( \vec{c} \)

Given \( \vec{a} = \vec{b} \times \vec{c} \), we have:

\(|\vec{a}| = |\vec{b}||\vec{c}||\sin \theta|,\)

where \( \theta \) is the angle between \( \vec{b} \) and \( \vec{c} \). Substituting \( |\vec{a}| = \sqrt{11}, |\vec{b}| = \sqrt{50}, |\vec{c}| = \sqrt{22} \):

\(\sqrt{11} = \sqrt{50} \cdot \sqrt{22} \cdot \sin \theta \quad \Rightarrow \quad \sin \theta = \frac{\sqrt{11}}{\sqrt{50} \cdot \sqrt{22}} = \frac{\sqrt{11}}{\sqrt{1100}} = \frac{\sqrt{11}}{10\sqrt{11}} = \frac{1}{10}.\)

Step 3: Magnitude of \( |\vec{b} + \vec{c}|^2 \)

\(|\vec{b} + \vec{c}|^2 = |\vec{b}|^2 + |\vec{c}|^2 + 2|\vec{b}||\vec{c}| \cos \theta,\)

where \( \cos \theta = \sqrt{1 - \sin^2 \theta} = \sqrt{1 - \left(\frac{1}{10}\right)^2} = \sqrt{1 - \frac{1}{100}} = \sqrt{\frac{99}{100}} = \frac{\sqrt{99}}{10} = \frac{3\sqrt{11}}{10}.\)

Substitute \( |\vec{b}|^2 = 50 \) and \( |\vec{c}|^2 = 22 \):

\(|\vec{b} + \vec{c}|^2 = 50 + 22 + 2 \cdot \sqrt{50} \cdot \sqrt{22} \cdot \frac{3\sqrt{11}}{10}.\)

Step 4: Simplify

\(|\vec{b} + \vec{c}|^2 = 72 + 2 \cdot \sqrt{1100} \cdot \frac{3\sqrt{11}}{10} = 72 + 2 \cdot 10\sqrt{11} \cdot \frac{3\sqrt{11}}{10} = 72 + 2 \cdot 3 \cdot 11 = 72 + 66 = 138.\)

Step 5: Calculate \( \left| 72 - |\vec{b} + \vec{c}|^2 \right| \)

\(\left| 72 - 138 \right| = \left| -66 \right| = 66.\)

Final Answer:

\(\left| 72 - |\vec{b} + \vec{c}|^2 \right| = 66.\)